14 LOCALLY CONVEX SPACES 65 exists a continuous mapping gmn of X into C, with support contained in Ow+1 (and therefore compact) and agreeing with hmn on On. Consequently, for each e > 0, each/e ^c(X) and each integer «, there exists an integer m such that pn(f— gmn) g £. Since the sequence of seminorms (/?„) is increasing, this proves the result, having regard to (12.14.3.1) and the definition of the topo- logy on #CPQ. Remark (12.14.6.3) The proof shows also that if Supp(/) c Un, then/is the limit of a sequence of functions gmn with supports contained in Un+1. (12.14.7) Let (/?a)a 61 be a family of seminorms on a vector space E. For each finite subset H of I, the function j7H = sup pa is a seminorm on E (12.14.1). aeH Since the set W((af); r) defined in (12.14.3.1) is the set of all x e E such that /?H(JC) < r (where H is the Set of the af), it follows that the family of seminorms pH, where H runs through all finite subsets of I, is equivalent to the family (pa). A set F of seminorms on E is said to be directed if for each pair of semi- norms p', p" belonging to F, there exists p e T such that p g: sup(p', p"). It follows that the topology of a locally convex space can always be defined by a directed set of seminorms. Let E be a locally convex space whose topology is defined by a set F of seminorms. If F is a vector subspace of E, it is clear that the induced topology on F is defined by the restrictions to F of the seminorms belonging to F. The corresponding result for quotient spaces is the following: (12.14.8) Let E be a vector space, F a vector subspace o/E, and n:E-+ E/F the canonical homomorphism. (i) Let p be a seminorm on E. For each x e E/F let (12.14.8.1) p(*) = inf p(z). Then p is a semi-norm on E/F. (ii) If F is a directed set of seminorms and ifE is endowed with the topology defined by F, then the quotient topology on E/F (12.11) is defined by the set of seminorms /), where p e F. (i) We have i = inf p(Az) « |A| inf p(z) == so that it remains to prove the triangle inequality. If x, y are elements of E/F there exists a dense